Communication and dominance in matrices

Communication and dominance matrices are used to represent relationships. Communication matrices capture the links or interactions between an individual or group, while dominance matrices represent hierarchical relationships among them.

Both are square matrices, as each row and column represent an individual or group. They are usually binary matrices, with 1 indicating a connection and 0 indicating no connection.

Use this page to revise the following concepts:

Communication Matrices

Communication matrices describe the interactions between individuals or groups and are typically symmetric. The main diagonal is usually 0, unless self-communication is considered.

Worked Example

For example, consider a group of four students (A, B, C, and D) working on a project. The communication matrix \(C\) represents whether two members communicate directly. In this case, the matrix is:

\(\space\space\space\space\space\space\space\space\space\space\space\space\textrm{A}\space \textrm{B}\space \textrm{C}\space \textrm{D}\)

\(C\) \(=\) \(\begin{align}& A\\& B\\& C\\& D\end{align}\) \(\left[\begin{align}& 0\space\space 1\space\space 1\space\space 0\\& 1\space\space 0\space\space 1\space\space 1\\& 1\space\space 1\space\space 0\space\space 1\\& 0\space\space 1\space\space 1\space\space 0\end{align}\right]\)

By inspecting the first row:

\(\space\space\space\textrm{A}\space \textrm{B}\space \textrm{C}\space \textrm{D}\)

\(A\left[0\space\space\space 1\space\space\space 1\space\space\space 0\right]\)

We can see that Student A directly communicates with B and C (one-step communication), but not D.

However, Student A can still communicate with D indirectly, through B or C, as they both directly communicate with D. This is an example of two-step communication.

To find all the two-step communications, we can calculate \(C^{2}\). The resulting matrix gives us the number of two-step communications between the students.

\(\space\space\space\space\space\space\space\space\space\space\space\space\textrm{A}\space \textrm{B}\space \textrm{C}\space \textrm{D}\)

\(C^{2}\) \(=\) \(\begin{align}& A\\& B\\& C\\& D\end{align}\) \(\left[\begin{align}& 2\space\space 1\space\space 1\space\space 2\\& 1\space\space 3\space\space 2\space\space 1\\& 1\space\space 2\space\space 3\space\space 1\\& 2\space\space 1\space\space 1\space\space 2\end{align}\right]\)

We can also determine the total number of one-step and two-step communications between the students by summing the matrices:

\(\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\textrm{A}\space \textrm{B}\space \textrm{C}\space \textrm{D}\)

\(C+C^{2}\) \(=\) \(\left[\begin{align}& 0\space\space 1\space\space 1\space\space 0\\& 1\space\space 0\space\space 1\space\space 1\\& 1\space\space 1\space\space 0\space\space 1\\& 0\space\space 1\space\space 1\space\space 0\end{align}\right]\) \(+\) \(\left[\begin{align}& 2\space\space 1\space\space 1\space\space 2\\& 1\space\space 3\space\space 2\space\space 1\\& 1\space\space 2\space\space 3\space\space 1\\& 2\space\space 1\space\space 1\space\space 2\end{align}\right]\) \(=\) \(\begin{align}& A\\& B\\& C\\& D\end{align}\) \(\left[\begin{align}& 2\space\space 2\space\space 2\space\space 2\\& 2\space\space 3\space\space 3\space\space 2\\& 2\space\space 3\space\space 3\space\space 2\\& 2\space\space 2\space\space 2\space\space 2\end{align}\right]\)


Dominance Matrices

Dominance matrices illustrate the hierarchical relationships between individuals or groups. These matrices are typically asymmetrical and binary, as dominance is a one-way relationship — when one individual dominates another, it occurs only once. The main diagonal is also always 0 since individuals cannot dominate themselves.

Worked Example

For example, in a round-robin competition with four players (A, B, C, and D), the dominance matrix \(D\) represents which players dominate (win against) the others. If the results of the competition were:

  • A beats B and C, but loses to D

  • B beats C and D, but loses to A

  • C beats D, but loses to A and B

  • D beats A and B, but loses to C

    • \(\space\space\space\space\space\space\space\space\space\space\space\space\textrm{A}\space \textrm{B}\space \textrm{C}\space \textrm{D}\)

    \(D\) \(=\) \(\begin{align}& A\\& B\\& C\\& D\end{align}\) \(\left[\begin{align}& 0\space\space 1\space\space 1\space\space 0\\& 0\space\space 0\space\space 1\space\space 1\\& 0\space\space 0\space\space 0\space\space 1\\& 1\space\space 1\space\space 0\space\space 0\end{align}\right]\)

    The total one-step dominance can be determined by summing the rows of \(D\):

  • A with 2 points

  • B with 2 points

  • C with 1 point

  • D with 2 points.

    • This suggests a tie between three players (A, B, and D). This is where dominance matrices prove useful. By examining the two-step dominance between players (while ignoring the main diagonal, as self-dominance does not make sense) we can further assess the situation.

    \(\space\space\space\space\space\space\space\space\space\space\space\space\textrm{A}\space \textrm{B}\space \textrm{C}\space \textrm{D}\)

    \(D^{2}\) \(=\) \(\begin{align}& A\\& B\\& C\\& D\end{align}\) \(\left[\begin{align}& 0\space\space 0\space\space 1\space\space 2\\& 1\space\space 1\space\space 0\space\space 1\\& 1\space\space 1\space\space 0\space\space 0\\& 0\space\space 1\space\space 2\space\space 1\end{align}\right]\)

    The total two-step dominance can be determined by summing the rows of \(D^{2}\):

  • A with 3 points

  • B with 3 points

  • C with 2 points

  • D with 4 points.

    • Adding the one-step dominance to the two-step dominance, we get the following totals:

  • A with 5 points

  • B with 5 points

  • C with 3 points

  • D with 6 points

    • Therefore, Player D would be considered the overall winner of the competition, having the highest total dominance.